A space elevator would be great for asteroid mining. It would have to be mounted at the equator and be attached at the top to a geosynchronous satellite.

The biggest problem is finding a strong enough cable. The deepest individual mine shaft on Earth goes down a little under 3 kilometers. The limiting factor is the strength of the cable pulling the car or ore skip to the surface. It must be enough to carry its own weight plus the weight of the payload. In deep shafts the weight of the cable far exceeds the weight of the payload. There are a few mines which go deeper than 3 kilometers, but they have to use subsidiary shafts. The space Machine Room Elevators Supplierselevator is in a completely different league. It would need a cable 36,000 kilometers long. If the counterweight design is used - it would be even longer. Even for the 3 Km shaft the cable needs to be tapered, so that the top which carries the largest load is thicker and stronger than the lower parts. A space elevator cable would need to be made of very exotic material, like boron nitride nanotubes and would have extreme tapering. The greatest stress is at geosynchronous orbit where the cable would need to be thickest. The issues are discussed at The Audacious Space Elevator. The consequences of an accident would be extreme.

The Japanese trial referenced in the comment with this question involves a steel cable only 10 meters long! A real space elevator will require a cable 3.6 million tiimes longer.